It seems to me that Benjen may be a White Walker.
His squad was ambushed.
Some of his squad were turned into wights.
Wights are only created by White Walkers.
At the end of season 2 we can see that the White Walker not only shares his same chin strap goatee, but also his hair and hairline.
I know this is all circumstantial but do you all think this could be a possibility?
In Season 6 Episode 6 of Game of Thrones:
Benjen Stark returns as a cloaked man on horseback (similar to Coldhands from the books) to save Bran and Meera.
In summary:
Benjen confirms he took a ranging party to find white walkers, but they found them first. A white walker stabbed him with a sword of ice, and he began to change, but the children of the forest found him and stopped the change by putting obsidian into his heart.
Here is a picture from the show:
And another with a nod to the book character's hands:
Based on the story evidence. He was most likely turned into a Wight not a White Walker.
It is speculated that White Walkers are created from human babies, which is gossip provided in the book by the rangers as to why Craster sacrifices his male children to the White Walkers. You can read more about that in this answer.
Other than that, we don't actually know of his fate since one of the purposes of The Great Ranging was to find out what happened to Benjen, but after they were annihilated at the Battle of the Fist of the First Men (what you saw at the end of Season 2 which was glossed over in the TV series), they have to retreat and as a result, we still don't know what happened to Benjen.
To add onto the correct answer by @Justin Ethier Benjen was neither wight nor White Walker but some other undead being saved from death/becoming a wight by the Children of the Forest.
BRAN: Uncle Benjen. The last letter Jon wrote me said you had been lost beyond the Wall.
BENJEN: I led a ranging party deep into the North to find White Walkers. They found us. A white walker stabbed me in the gut with a sword of ice. Left me there to die. To turn. The Children found me. Stopped the walker’s magic from taking hold.
BRAN: How?
BENJEN: The same way they made the Walkers in the first place. You saw it yourself.
BRAN: Dragonglass. A shard of dragonglass plunged into your heart.
Game of Thrones, S06 E06, "Blood of my Blood"
However, as of Season 7 Episode 6, "Beyond the Wall", he is presumed truly dead after sacrificing himself for Jon.
Go! Go now! Leave! Uncle Benjen! How? You ride for the pass.
Come with me.
It's not time.
Game of Thrones, S07 E06, "Beyond the Wall"
You can watch the scene here where he gets taken down by an overwhelming majority of wights:
